Wonderful Compactification Research Articles

The distribution of integral points on the wonderful compactification by height

The distribution of integral points on the wonderful compactification by height

Kähler–Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII

The wonderful compactification [Formula: see text] of a symmetric homogeneous space of type AIII[Formula: see text] for each [Formula: see text] is Fano, and its blowup [Formula: see text] along the unique closed orbit is Fano if [Formula: see text] and Calabi–Yau if [Formula: see text]. Using a combinatorial criterion for K-polystability of smooth Fano spherical varieties obtained by Delcroix, we prove that [Formula: see text] admits a Kähler–Einstein metric for each [Formula: see text] and [Formula: see text] admits a Kähler–Einstein metric if and only if [Formula: see text].

Wonderful compactifications and rational curves with cyclic action

Abstract We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows.

The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}//(\pmb{\mathbb{C}}^{\ast})^n$ of the Grassmann manifolds $G_{n,2}$

The complex Grassmann manifolds $G_{n,k}$ appear as one of the fundamental objects in developing an interaction between algebraic geometry and algebraic topology. The case $k=2$ is of special interest on its own as the manifolds $G_{n,2}$ have several remarkable properties which distinguish them from the $G_{n,k}$ for $k>2$. In our paper we obtain results which, essentially using the specifics of the Grassmann manifolds $G_{n,2}$, develop connections between algebraic geometry and equivariant topology. They are related to well-known problems of the canonical action of the algebraic torus $(\mathbb{C}^{\ast})^n$ on $G_{n,2}$ and the induced action of the compact torus $T^n\subset(\mathbb{C}^{\ast})^n$. Kapranov proved that the Deligne-Mumford-Grothendieck-Knudsen compactification $\overline{\mathcal{M}}(0,n)$ of the space of $n$-pointed rational stable curves can be realized as the Chow quotient $G_{n,2}//(\mathbb{C}^{\ast})^{n}$. In recent papers of the authors a constructive description of the orbit space $G_{n,2}/T^n$ was obtained. In deducing this result the notions of the complex of admissible polytopes and the universal space of parameters $\mathcal{F}_{n}$ for the $T^n$-action on $G_{n,2}$ were of essential use. Using the techniques of wonderful compactification, in this paper an explicit construction of the space $\mathcal{F}_{n}$ is presented. In combination with Keel's description of $\overline{\mathcal{M}}(0,n)$, this construction enabled one to obtain an explicit diffeomorphism between $\mathcal{F}_{n}$ and $\overline{\mathcal{M}}(0,n)$. In this way, we give a description of $G_{n,2}//(\mathbb{C}^{\ast})^{n}$ as the space $\mathcal{F}_{n}$ with a structure described in terms of admissible polytopes $P_\sigma$ and spaces $F_\sigma$. Bibliography: 32 titles.

Wonderful asymptotics of matrix coefficient D-modules

Beilinson–Bernstein localization realizes representations of complex reductive Lie algebras as monodromic D-modules on the “basic affine space” G/N, a torus bundle over the flag variety. A doubled version of the same space appears as the horocycle space describing the geometry of the reductive group G at infinity, near the closed stratum of the wonderful compactification G‾, or equivalently in the special fiber of the Vinberg semigroup of G. We show that Beilinson–Bernstein localization for Ug-bimodules arises naturally as the specialization at infinity in G‾ of the D-modules on G describing matrix coefficients of Lie algebra representations. More generally, the asymptotics of matrix coefficient D-modules along any stratum of G‾ are given by the matrix coefficient D-modules for parabolic restrictions. This provides a simple algebraic derivation of the relation between growth of matrix coefficients of admissible representations and n-homology. The result is an elementary consequence of the compatibility of localization with the degeneration of affine G-varieties to their asymptotic cones; analogous results hold for the asymptotics of the equations describing spherical functions on symmetric spaces.

Steinberg slices and group-valued moment maps

We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer Z of G, which is equipped with the usual symplectic structure in this way. We construct a smooth relative compactification Z‾ by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this relative compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.

Complete symplectic quadrics and Kontsevich spaces of conics in Lagrangian Grassmannians

A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal crossing. We construct the wonderful compactification of the space of symmetric and symplectic matrices, and investigate its geometry. As an application, we describe the birational geometry of the Kontsevich spaces parametrizing conics in Lagrangian Grassmannians.

Complete moduli of cubic threefolds and their intermediate Jacobians

The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the moduli space of principally polarized abelian fivefolds. A much better "wonderful" compactification of the space of cubic threefolds was constructed by the first and fourth authors --- it has a modular interpretation, and divisorial normal crossing boundary. We prove that the intermediate Jacobian map extends to a morphism from the wonderful compactification to the second Voronoi toroidal compactification of the moduli of principally polarized abelian fivefolds --- the first and fourth author previously showed that it extends to the Satake compactification. Since the second Voronoi compactification has a modular interpretation, our extended intermediate Jacobian map encodes all of the geometric information about the degenerations of intermediate Jacobians, and allows for the study of the geometry of cubic threefolds via degeneration techniques. As one application we give a complete classification of all degenerations of intermediate Jacobians of cubic threefolds of torus rank 1 and 2.

Generalized moment graphs and the equivariant intersection cohomology of BXB-orbit closures in the wonderful compactification of a group

We provide a functorial description of the torus equivariant intersection cohomology of the Borel orbit closures in the wonderful compactification of a semisimple adjoint complex algebraic group. Ours is an adaptation of the moment graph approach of Goresky, Kottwitz, and MacPherson. We first define a generalized moment graph, a combinatorial object determined by the structure of a finite collection of low-dimensional torus orbits. We then show how an algorithm of Braden and MacPherson computes the stalks of the equivariant intersection cohomology complex, producing “sheaves” on the generalized moment graph that functorially encode the equivariant intersection cohomology. Because the Braden-MacPherson algorithm uses only basic commutative algebra, this approach greatly simplifies the a priori very difficult task of computing intersection cohomology stalks. This paper extends previous work of Springer, who had computed the (ordinary) intersection cohomology Betti numbers of the Borel orbit closures in the wonderful compactification.

Wonderful compactification of character varieties

Using the wonderful compactification of a semisimple adjoint affine algebraic group G defined over an algebraically closed field k of arbitrary characteristic, we construct a natural compactification Y of the G-character variety of any finitely generated group F. When F is a free group, we show that this compactification is always simply connected with respect to the \'etale fundamental group, and when k=C it is also topologically simply connected. For other groups F, we describe conditions for the compactification of the moduli space to be simply connected and give examples when these conditions are satisfied, including closed surface groups and free abelian groups when G=PGL(n,C). Additionally, when F is a free group we identify the boundary divisors of Y in terms of previously studied moduli spaces, and we construct a family of Poisson structures on Y and its boundary divisors arising from Belavin-Drinfeld splittings of the double of the Lie algebra of G. In the appendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson structure on a quotient of a Poisson algebraic variety by the action of a reductive Poisson algebraic group.

Divisors on matroids and their volumes

The classical volume polynomial in algebraic geometry measures the degrees of ample (and nef) divisors on a smooth projective variety. We introduce an analogous volume polynomial for matroids, and give a complete combinatorial formula. For a realizable matroid, we thus obtain an explicit formula for the classical volume polynomial of the associated wonderful compactification. We then introduce a new invariant called the shifted rank volume of a matroid as a particular specialization of its volume polynomial, and discuss its algebro-geometric and combinatorial properties in connection to linear series on blow-ups of projective spaces.

Bethe Subalgebras in Yangians and the Wonderful Compactification

We study the family of Bethe subalgebras in the Yangian $Y(\mathfrak{g})$ parameterized by the corresponding adjoint Lie group $G$. We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group $G_1[[t^{-1}]]$. In particular we show that, for regular values of the parameter, these subalgebras are free polynomial algebras with the same Poincare series as the Cartan subalgebra of the Yangian. Next, we extend the family of Bethe subalgebras to the De Concini--Procesi wonderful compactification $\overline{G}\supset G$ and describe the subalgebras corresponding to generic points of any stratum in $\overline{G}$ as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in $\mathfrak{g}$.

On endomorphisms of arrangement complements

Let $\Omega$ be the complement of a connected, essential hyperplane arrangement. We prove that every dominant endomorphism of $\Omega$ extends to an endomorphism of the tropical compactification $X$ of $\Omega$ associated to the Bergman fan structure on the tropicalization of $\Omega$. This generalizes a previous result by Remy, Thuillier and the second author which states that every automorphism of Drinfeld's half-space over a finite field $\mathbb{F}_q$ extends to an automorphism of the successive blow-up of projective space at all $\mathbb{F}_q$-rational linear subspaces. This successive blow-up is in fact the minimal wonderful compactification by de Concini and Procesi, which coincides with $X$ by results of Feichtner and Sturmfels. Whereas the previous proof is based on Berkovich analytic geometry over the trivially valued finite ground field, the generalization discussed in the present paper relies on matroids and tropical geometry.

Ricci flat Kähler metrics on rank two complex symmetric spaces

We obtain Ricci flat K\"ahler metrics on complex symmetric spaces of rank two by using an explicit asymptotic model whose geometry at infinity is interpreted in the wonderful compactification of the symmetric space. We recover the metrics of Biquard-Gauduchon in the Hermitian case and obtain in addition several new metrics.

The wonderful compactification for quantum groups

In this paper, we introduce a quantum version of the wonderful compactification of a group as a certain noncommutative projective scheme. Our approach stems from the fact that the wonderful compactification encodes the asymptotics of matrix coefficients, and from its realization as a GIT quotient of the Vinberg semigroup. In order to define the wonderful compactification for a quantum group, we adopt a generalized formalism of $\mathsf{Proj}$ categories in the spirit of Artin and Zhang. Key to our construction is a quantum version of the Vinberg semigroup, which we define as a $q$-deformation of a certain Rees algebra, compatible with a standard Poisson structure. Furthermore, we discuss quantum analogues of the stratification of the wonderful compactification by orbits for a certain group action, and provide explicit computations in the case of $\mathrm{SL}_2$.

On an invariant bilinear form on the space of automorphic forms via asymptotics

This article concerns the study of a new invariant bilinear form $\mathcal B$ on the space of automorphic forms of a split reductive group $G$ over a function field. We define $\mathcal B$ using the asymptotics maps from Bezrukavnikov-Kazhdan and Sakellaridis-Venkatesh, which involve the geometry of the wonderful compactification of $G$. We show that $\mathcal B$ is naturally related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary. In the proof, we highlight the connection between the classical non-Archimedean Gindikin-Karpelevich formula and certain factorization algebras acting on geometric Eisenstein series. We then give another definition of $\mathcal B$ using the constant term operator and the inverse of the standard intertwining operator. The form $\mathcal B$ defines an invertible operator $L$ from the space of compactly supported automorphic forms to a new space of "pseudo-compactly" supported automorphic forms. We give a formula for $L^{-1}$ in terms of pseudo-Eisenstein series and constant term operators which suggests that $L^{-1}$ is an analog of the Aubert-Zelevinsky involution.

Deligne–Lusztig duality and wonderful compactification

We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne–Lusztig (or Alvis–Curtis) duality for p-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for $$G=GL(n)$$ and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a p-adic group.

Wonderful compactifications of Bruhat-Tits buildings

Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.

Wonderful compactifications of the moduli space of points in affine and projective space

We introduce and study smooth compactifications of the moduli space of n labeled points with weights in projective space, which have normal crossings boundary and are defined as GIT quotients of the weighted Fulton–MacPherson compactification. We show that the GIT quotient of a wonderful compactification is also a wonderful compactification under certain hypotheses. We also study a weighted version of the configuration spaces parametrizing n points in affine space up to translation and homothety. In dimension one, the above compactifications are isomorphic to Hassett’s moduli space of rational weighted stable curves.

The Peterson variety and the wonderful compactification

We look at the centralizer in a semisimple algebraic group G G of a regular nilpotent element e ∈ Lie ( G ) e\in \text {Lie}(G) and show that its closure in the wonderful compactification is isomorphic to the Peterson variety. It follows that the closure in the wonderful compactification of the centralizer G x G^x of any regular element x ∈ Lie ( G ) x\in \text {Lie}(G) is isomorphic to the closure of a general G x G^x -orbit in the flag variety. We also give a description of the G e G^e -orbit structure of the Peterson variety.